Problem: All of the 4th grade teachers and students from Almond went on a field trip to an art museum. Tickets were $$8.50$ each for teachers and $$4.50$ each for students, and the group paid $$48.00$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$25.50$ each for teachers and $$11.00$ each for students, and the group paid $$131.50$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+4.5y = 48}$ ${25.5x+11y = 131.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-25.5x-13.5y = -144}$ ${25.5x+11y = 131.5}$ Add the top and bottom equations together. $ -2.5y = -12.5 $ $ y = \dfrac{-12.5}{-2.5}$ ${y = 5}$ Now that you know ${y = 5}$ , plug it back into $ {8.5x+4.5y = 48}$ to find $x$ ${8.5x + 4.5}{(5)}{= 48}$ $8.5x+22.5 = 48$ $8.5x = 25.5$ $x = \dfrac{25.5}{8.5}$ ${x = 3}$ You can also plug ${y = 5}$ into $ {25.5x+11y = 131.5}$ and get the same answer for $x$ ${25.5x + 11}{(5)}{= 131.5}$ ${x = 3}$ There were $3$ teachers and $5$ students on the field trips.